AERODYNAMICS

Corresponding Member of the Academy of Sciences of the USSR G. G. CHERNYI

A FLAT WING IN A HYPERSONIC FLOW

Consider the hypersonic flow of an ideal gas past a flat wing at an angle of attack. To describe the motion of the gas on the windward side of the wing, we introduce Cartesian coordinates: the axes \(x\) and \(y\) are placed in the plane of the wing, and the axis \(z\) is directed along the normal to this plane.

We take the equations of motion of the gas in the layer between the wing surface and the shock wave in the following form:

\[ \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} =0, \qquad \frac{\partial}{\partial x}(\rho u^2+p) + \frac{\partial \rho uv}{\partial y} + \frac{\partial \rho uw}{\partial z} =0, \]

\[ \frac{\partial \rho vu}{\partial x} + \frac{\partial}{\partial y}(\rho v^2+p) + \frac{\partial \rho vw}{\partial z} =0, \]

\[ \frac{\partial \rho uS}{\partial x} + \frac{\partial \rho vS}{\partial y} + \frac{\partial \rho wS}{\partial z} =0, \qquad \frac{\partial \rho ui^*}{\partial x} + \frac{\partial \rho vi^*}{\partial y} + \frac{\partial \rho wi^*}{\partial z} =0. \tag{1} \]

Here \(u, v, w\) are the velocity components along the axes \(x, y, z\); \(\rho, p, S, i^*\) denote the density, pressure, entropy, and total heat content per unit mass of gas. For a perfect gas with constant specific heats,

\[ S=\frac{p^{1/\gamma}}{\rho},\qquad i^*=\frac{u^2+v^2+w^2}{2}+\frac{\gamma}{\gamma-1}\frac{p}{\rho}. \]

The system of equations (1) can be rewritten in the general form

\[ \frac{\partial A_{ij}}{\partial x_j} + \frac{\partial B_i}{\partial z} =0; \tag{2} \]

where \(x_1=x,\ x_2=y;\ A_{11}=\rho u,\ A_{12}=\rho v,\ B_1=\rho w,\ A_{21}=\rho u^2+p,\ A_{22}=\rho uv,\ B_2=\rho uw;\ A_{31}=\rho vu,\ A_{32}=\rho v^2+p;\ B_3=\rho vw,\ A_{41}=\rho uS,\ A_{42}=\rho vS,\ B_4=\rho wS,\ A_{51}=\rho ui^*,\ A_{52}=\rho vi^*,\ B_5=\rho wi^*\).

Let \(h=h(x,y)\) be the thickness of the gas layer between the wing surface and the shock wave. Introduce, instead of the coordinate \(z\), the variable \(\zeta\) by the formula \(\zeta=2z/h-1\); multiply term by term the equations of the system (2) by the Legendre polynomial of order \(n\), \(P_n(\zeta)\), and integrate them with respect to \(z\) from \(0\) to \(h\).

Carrying out simple transformations with the use of the properties of Legendre polynomials and of the conditions \(\hat B_i(x_1,x_2,0)=0\), which follow from the condition \(w=0\) at \(z=0\), we obtain, as a result, for \(n=1,2,3,\ldots\)

\[ \frac{\partial}{\partial x_j}\, \frac{h}{2} \int_{-1}^{1} P_n A_{ij}\,d\zeta + \frac{1}{2} \frac{\partial h}{\partial x_j} \int_{-1}^{1} \left[ nP_n+ \sum_{k=1}^{n}(2n-2k+1)P_{n-k} \right] A_{ij}\,d\zeta - \]

\[ - \int_{-1}^{1} B_i \sum_{k=1}^{N} (2n-4k+3)P_{n-2k+1}\,d\zeta + \left( B_i-A_{ij}\frac{\partial h}{\partial x_j} \right)_{z=h} =0, \]

\[ N=n/2\ \text{or}\ (n+1)/2 \quad \text{(depending on whether \(n\) is even or odd);} \quad \text{for } n=0 \]

\[ \frac{\partial}{\partial x_j}\, \frac{h}{2} \int_{-1}^{1} A_{ij}\,d\zeta + \left( B_i-A_{ij}\frac{\partial h}{\partial x_j} \right)_{z=h} =0. \]

The conditions of conservation of mass, momentum (in projections on the axes \(x\) and \(y\)), and energy at the shock wave have the form:

\[ \left( B_i-A_{ij}\frac{\partial h}{\partial x_j} \right)_{z=h} = B^\infty - A_{ij}^{\infty}\frac{\partial h}{\partial x_j}, \qquad i=1,2,3,5. \]

Quantities in the incident flow are denoted by the index \(\infty\). If, by means of these conditions and the condition of conservation of momentum in the projection on the \(z\)-axis, the value of the entropy behind the shock wave \(S_h\) is expressed through the parameters of the incident flow and \(\partial h/\partial x_j\), then one may write

\[ \left(B_4-A_{4j}\frac{\partial h}{\partial x_j}\right)_{z=h} = B_4^\infty-A_{4j}^\infty\frac{\partial h}{\partial x_j}, \]

where \(B_4^\infty=B_1^\infty S_h,\ A_{4j}^\infty=A_{1j}^\infty S_h\). For a perfect gas with constant specific heats,

\[ S_h=\frac{1}{\rho^\infty} \left[ \frac{2}{(\gamma+1)\rho^\infty} \frac{\left(B_1^\infty-A_{1j}^\infty\dfrac{\partial h}{\partial x_j}\right)^2} {1+\dfrac{\partial h}{\partial x_j}\dfrac{\partial h}{\partial x_j}} -\frac{\gamma-1}{\gamma+1}p^\infty \right]^{1/\gamma} \times \]

\[ \times \left[ \frac{\gamma-1}{\gamma+1} + \frac{2\gamma}{\gamma+1}p^\infty\rho^\infty \frac{ 1+\dfrac{\partial h}{\partial x_j}\dfrac{\partial h}{\partial x_j} }{ \left(B_1^\infty-A_{1j}^\infty\dfrac{\partial h}{\partial x_j}\right)^2 } \right]. \tag{3} \]

The equation of conservation of the total heat content can subsequently be replaced by the Bernoulli integral

\[ \frac{u^2+v^2+w^2}{2} + \frac{\gamma}{\gamma-1}\frac{p}{\rho} = \frac{V_\infty^2}{2} + \frac{\gamma}{\gamma-1}\frac{p^\infty}{\rho^\infty}. \]

Using what has been said, after certain transformations the system of equations can be rewritten in the following form:

\[ \frac{\partial}{\partial x_j}\frac{h}{2}\int_{-1}^{1}A_{ij}\,d\xi + B_i^\infty - A_{ij}^\infty\frac{\partial h}{\partial x_j} =0, \]

\[ \frac{\partial}{\partial x_j}\frac{h}{2}\int_{-1}^{1}(P_n-P_{n-1})A_{ij}\,d\xi + \frac{1}{2}\frac{\partial h}{\partial x_j}\int_{-1}^{1}n(P_n+P_{n-1})A_{ij}\,d\xi - \tag{4} \]

\[ - \int_{-1}^{1} B_i\sum_{k=0}^{n-1}(-1)^{\,n+k}(2k+1)P_k\,d\xi =0, \qquad n=1,2,3,\ldots;\quad i=1,2,3,4; \]

\[ \int_{-1}^{1} P_n \left[ \rho\left(\frac{u^2+v^2+w}{2}-i^\infty\right) + \frac{\gamma}{\gamma-1}p \right]d\xi. \]

Let us call the polynomial

\[ u^{(n)}=\sum_{k=0}^{n}P_k(\xi)u_k^{(n)}(x,y) \]

the \(n\)-th approximation for the function \(u\). Then, for \(n\ge 1\), to determine the \(5(n+1)\) coefficients of the \(n\)-th approximations \(u^{(n)}, v^{(n)}, w^{(n)}, p^{(n)}, \rho^{(n)}\) and the function \(h\), we have \(4n\) first-order differential equations of the system (4), \(n\) finite relations following from the Bernoulli integral, 4 relations on the shock wave (for \(i=1,2,3,4\)), and one finite relation

\[ w_0^{(n)}-w_1^{(n)}+w_2^{(n)}\ldots(-1)^n w_n^{(n)}=0, \]

which follows from the boundary condition \(w=0\) at \(z=0\).

The system of equations of the zeroth approximation \((n=0)\) has the following form (the indices on the unknown quantities are omitted):

\[ \frac{\partial\rho u h}{\partial x} + \frac{\partial\rho v h}{\partial y} + \rho^\infty V_\nu =0, \]

\[ \rho h\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right) + \frac{\partial}{\partial x}(p-p^\infty)h + \rho^\infty(U-u)V_\nu =0, \]

\[ \rho h\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right) + \frac{\partial}{\partial y}(p-p^\infty)h + \rho^\infty(V-v)V_\nu =0, \]

\[ \rho h\left(u\frac{\partial S}{\partial x}+v\frac{\partial S}{\partial y}\right) + \rho^\infty(S_h-S)V_\nu =0, \tag{5} \]

\[ \frac{u^2+v^2}{2} + \frac{\gamma}{\gamma-1}\frac{p}{\rho} = \frac{V_\infty^2}{2} + \frac{\gamma}{\gamma-1}\frac{p^\infty}{\rho^\infty}. \]

Here \(V_\nu = W - U\,\partial h/\partial x - V\,\partial h/\partial y\), and \(S_h\) is determined by formula (3).

To solve the system obtained, boundary conditions must be formulated on the wing contour. On that part of the contour where the shock wave is attached to the wing edge, \(h=0\), while the values of the remaining unknown functions are determined from the relations on the wave. The boundary conditions on the remaining part of the contour in the general case cannot be prescribed in advance, since the flow on the windward side of the wing and the flow on its leeward side must be calculated simultaneously.

For sufficiently large angles of attack of the wing and Mach number of the oncoming flow, when the pressure on the windward side of the wing is much greater than on the leeward side, on that part of the contour where \(h \ne 0\) and the specification of a boundary condition is necessary, such a condition will be the equality of the normal component of the gas velocity to the local speed of sound.

The solution of system (5) can be carried out by methods analogous to those used in solving the usual problems of two-dimensional steady gas flows (it must be taken into account here that the entropy equation is essentially nonlinear). In particular, at supersonic speeds the method of characteristics can be used to solve system (5). System (5) has the following families of characteristics:

\[ y_{1,2}'=(uv'\pm a\sqrt{u^2+v^2-a^2})/(u^2-a^2),\qquad y_3'=v/u,\qquad y_4'=\Phi_{h_y}'/\Phi_{h_x}' \]

(the characteristics of the third family are doubled). Along the characteristics the following relations must be satisfied, respectively:

\[ v' + \frac{y'}{y_1'y_2'}\,u' = \mp \frac{a\sqrt{u^2+v^2-a^2}}{v^2-a^2}\,y'\Omega + \]

\[ + \frac{a^2y'}{\rho h\,(v^2-a^2)} \left( \rho u h_x+\rho v h_y+ \frac{\rho^\infty V_\nu}{\gamma-1}\, \frac{\gamma S_h-S}{S} \right), \]

\[ \rho h u S' + \rho^\infty (S_h-S)V_\nu=0, \]

\[ \Phi_{h_x}'h_y'= -\Phi_u'\frac{\partial u}{\partial y} -\Phi_v'\frac{\partial v}{\partial y} -\Phi_\rho'\frac{\partial \rho}{\partial y} -\Phi_S'\frac{\partial S}{\partial y}. \]

Here

\[ \Omega\equiv \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} = -\frac{1}{\rho u h} \left[ (p-p^\infty)h_y+\rho^\infty (V-v)V_\nu -\frac{\gamma}{\gamma-1}\frac{ph}{S}\frac{\partial S}{\partial y} \right], \]

\[ \Phi\equiv (p-p^\infty)(uh_x+vh_y) +\rho^\infty \left[ Uu+Vv-u^2-v^2+ \frac{\gamma}{\gamma-1}p^{(\gamma-1)/\gamma}(S_h-S) \right]V_\nu; \]

the quantities

\[ \frac{\partial u}{\partial y},\quad \frac{\partial v}{\partial y},\quad \frac{\partial \rho}{\partial y},\quad \frac{\partial S}{\partial y} \]

are easily expressed in terms of the corresponding values \(u'\), \(v'\), \(\rho'\), \(S'\); for example,

\[ \rho h\,(v-uy')\frac{\partial S}{\partial y} = -\rho h u S' -\rho^\infty(S_h-S)V_\nu . \]

The first three families of characteristics are the acoustic characteristics and streamlines customary for two-dimensional gas-dynamic problems; the fourth family is new and has no analogue in ordinary two-dimensional gas-dynamic problems.

Thus, the problem of determining the three-dimensional flow on the windward side of a flat wing at large angles of attack and large Mach numbers is approximately reduced to determining a two-dimensional flow in the plane of the wing by means of the system of equations (5), with the corresponding boundary conditions on the wing contour.

Interesting results have been obtained in the analysis of solutions of system (5) for conical flows. These results have been used to analyze the flow about triangular wings. In Fig. 1, referring to the case \(M=\infty\) and \(\gamma=1.4\), lines are plotted that separate the regions of different types of symmetric flow about a triangular wing. In regions \(I\)–\(IV\) conical flow regimes are possible. In this case, to each pair of values of the angle of attack \(\alpha\) and the semi-apex angle of the wing \(\theta_0\) there correspond two solutions, as well as

as in the flow past a flat wing of infinite span \((\theta_0=\pi/2)\). In regions \(V\)—\(VII\) there are no conical flows (the exception is the intersection of regions \(VI\) and \(IV\)). In region \(I\) and in part of regions \(II\) and \(III\) to the left of the dashed line the trailing edge of the wing has no influence upstream (in the solution with the weaker wave), and, consequently, the solutions obtained for an infinite wing are also suitable for a finite wing. In the remaining region, changes in the parameters must be considered for finite wings.

Fig. 1

For example, below in Fig. 1 the change in the flow pattern around a wing with \(\theta_0=10^\circ\) is shown as the angle of attack changes from \(0\) to \(180^\circ\).

In region \(I\) the shock is attached to the leading edges of the wing. Behind the shock near the edge there is a region of translational flow, passing continuously (through an acoustic characteristic) into a flow whose streamlines asymptotically approach the center line of the wing. On passing from region \(I\) to region \(II\), the shock becomes detached along the edges, but retains a common vertex with the wing. The velocity component normal to the wing edge is then equal to the speed of sound. The lines on which the circumferential component of the velocity vanishes and from which the flow spreads toward the wing edges and toward its center line approach one another as the angle of attack increases and, upon transition to region \(III\), merge with the center line (for wing semivertex angles greater than a certain \(\theta_0^*(M)\), this merging does not occur before the conical character of the flow is destroyed). The flow then appears as in Fig. 1, \(III\). Upon transition to region \(IV\), the flow on the center line of the wing changes direction: the gas begins to move toward the wing vertex. For a finite wing this means the appearance of a critical point on its surface. As the angle of attack increases and region \(V\) is reached, conical flow becomes impossible, and one must necessarily consider a wing of finite size. As the angle of attack grows further, the critical point shifts toward the wing vertex and merges with it; upon entering region \(VI\), the shock becomes attached along the trailing edge, and near it a region of translational flow appears, which gradually grows and, upon entering region \(VII\), fills the entire surface of the wing.

With some modifications, the described change in the types of flow also occurs for wings with other vertex angles.

The solution obtained has made it possible to find the distributions of all quantities of interest in regimes of conical flow.

In conclusion, we note that the derivation of system (5) presented in the present work refines and substantiates the approximate approach to the solution of the problem of hypersonic flow past a wing used earlier by the author in work (1).

Moscow State University
named after M. V. Lomonosov

Received
31 XII 1964

REFERENCES

1. G. G. Chernyi, DAN, 155, No. 2 (1964).